z-transform of a^{|k|}

Definition. a^{|k|} is a two-sided sequence (k from −∞ to ∞), so the bilateral transform is used:

X(z) = Σ_{k=−∞}^{∞} a^{|k|} z⁻ᵏ , taking |a| < 1.

Split into k ≥ 0 and k < 0:

X(z) = Σ_{k=0}^{∞} (a/z)ᵏ + Σ_{k=1}^{∞} (a z)ᵏ

• First series: converges for |z| > |a| to z/(z − a).
• Second series (let m = −k): converges for |z| < 1/|a| to az/(1 − az).

X(z) = z/(z − a) + az/(1 − az)
     = [ z(1 − az) + az(z − a) ] / [ (z − a)(1 − az) ]
     = z (1 − a²) / [ (z − a)(1 − az) ]

Answer: X(z) = z(1 − a²) / [(z − a)(1 − az)], region of convergence |a| < |z| < 1/|a|.

If only k ≥ 0 is considered, the transform is simply z/(z − a) with |z| > |a|. (Numerical check at a = 0.5, z = 1.2: both forms give 3.2143.)